||It has been suggested that this article be merged into Quantile. (Discuss) Proposed since February 2013.|
In statistics, a percentile (or a centile) is the value of a variable below which a certain percent of observations fall. For example, the 20th percentile is the value (or score) below which 20 percent of the observations may be found. The term percentile and the related term percentile rank are often used in the reporting of scores from norm-referenced tests. For example, if a score is in the 86th percentile, it is higher than 86% of the other scores.
Nearest rank 
rounding the result to the nearest integer, and then taking the value that corresponds to that rank. (Note that the rounded value of n is just the least integer which exceeds .)
For example, by this definition, given the numbers
- 15, 20, 35, 40, 50
the rank of the 30th percentile would be
Thus the 30th percentile is the second number in the sorted list, 20.
The 35th percentile would have rank
so the 35th percentile would be the second number again (since 2.25 rounds down to 2) or 20
The 40th percentile would have rank
so the 40th percentile would be the third number (since 2.5 rounds up to 3), or 35.
The 100th percentile is defined to be the largest value. (In this case we do not use the above definition with P=100, because the rank n would be greater than the number N of values in the original list.)
In lists with fewer than 100 values the same number can occupy more than one percentile group.
Linear interpolation between closest ranks 
An alternative to rounding used in many applications is to use linear interpolation between the two nearest ranks.
In particular, given the N sorted values , we define the percent rank corresponding to the nth value as:
In this way, for example, if then the percent rank corresponding to the third value is:
The value v of the P-th percentile may now be calculated as follows:
If or , then we take or , respectively.
If there is some integer k for which , then we take .
Otherwise, we find the integer k for which , and take
Using the list of numbers above, the 40th percentile would be found by linearly interpolating between the 30th percentile, 20, and the 50th, 35. Specifically:
This is halfway between 20 and 35, which one would expect since the rank was calculated above as 2.5.
It is readily confirmed that the 50th percentile of any list of values according to this definition of the P-th percentile is just the sample median.
Moreover, when N is even the 25th percentile according to this definition of the P-th percentile is the median of the first values (i.e., the median of the lower half of the data).
Weighted percentile 
In addition to the percentile function, there is also a weighted percentile, where the percentage in the total weight is counted instead of the total number. There is no standard function for a weighted percentile. One method extends the above approach in a natural way.
Suppose we have positive weights associated, respectively, with our N sorted sample values. Let
the -th partial sum of the weights. Then the formulas above are generalized by taking
The 50% weighted percentile is known as the weighted median.
Alternative methods 
Some software packages, including Microsoft Excel (up to the version 2007) use the following method, noted as an alternative by NIST to estimate the value, , of the P-th percentile of an ascending ordered dataset containing N elements with values .
The rank is calculated:
and then split into its integer component k and decimal component d, such that . Then is calculated as:
These two approaches give the rank of the 40th percentile in the above example as, respectively:
The values are then interpolated as usual based on these ranks, yielding 29 and 26, respectively, for the 40th percentile.
EXCEL Definitions Updated 
New functions have been introduced in EXCEL 2010. These are explained here. The PERCENTRANK is the same as PERCENTILERANK.INC, which maps the smallest number in list to 0%, the largest to 100%, and the ranks in the middle linearly, so 1,2,3,4 go to 0/3, 1/3, 2/3 and 3/3. There are three arguments to the function, the third argument is significance, which defaults to 2. You can increase (or decrease it) if you like. If default is used, then 1/3 comes out to 33.300%, which is baffling since it differs from the expected value of 33.333%
For a list of n objects, the PERCENTILERANK.EXC function maps the rank 0 (which is not part of the dataset) to 0%, and the rank n+1 to 1, so the numbers 1,2,3,4 end up at 1/5, 2/5, 3/5, and 4/5.
When ISPs bill "burstable" internet bandwidth, the 95th or 98th percentile usually cuts off the top 5% or 2% of bandwidth peaks in each month, and then bills at the nearest rate. In this way infrequent peaks are ignored, and the customer is charged in a fairer way. The reason this statistic is so useful in measuring data throughput is that it gives a very accurate picture of the cost of the bandwidth. The 95th percentile says that 95% of the time, the usage is below this amount. Just the same, the remaining 5% of the time, the usage is above that amount.
The 85th percentile speed of traffic on a road is often used as a guideline in setting speed limits and assessing whether such a limit is too high or low.
The normal curve and percentiles 
The methods given above are approximations for use in small-sample statistics. In general terms, for very large populations percentiles may often be represented by reference to a normal curve plot. The normal curve is plotted along an axis scaled to standard deviation, or sigma, units. Mathematically, the normal curve extends to negative infinity on the left and positive infinity on the right. Note, however, that a very small portion of individuals in a population will fall outside the −3 to +3 range.
In humans, for example, a small portion of all people can be expected to fall above the +3 sigma height level.
Percentiles represent the area under the normal curve, increasing from left to right. Each standard deviation represents a fixed percentile. Thus, rounding to two decimal places, −3 is the 0.13th percentile, −2 the 2.28th percentile, −1 the 15.87th percentile, 0 the 50th percentile (both the mean and median of the distribution), +1 the 84.13th percentile, +2 the 97.72nd percentile, and +3 the 99.87th percentile. Note that the 0th percentile falls at negative infinity and the 100th percentile at positive infinity.
See also 
- Hyndman RH, Fan Y (1996). "Sample quantiles in statistical packages". The American Statistician 50 (4): 361–365. doi:10.2307/2684934. JSTOR 2684934.
- Lane, David. "Percentiles". Retrieved 2007-09-15.
- Pottel, Hans. "Statistical flaws in Excel". Retrieved 2013-03-25.
- Schoonjans F, De Bacquer D, Schmid P (2011). "Estimation of population percentiles". Epidemiology 22 (5): 750–751. doi:10.1097/EDE.0b013e318225c1de.
- "Matlab Statistics Toolbox – Percentiles". Retrieved 2006-09-15., This is equivalent to Method 5 discussed here
- "Engineering Statistics Handbook: Percentile". NIST. Retrieved 2009-02-18.
- Free Online Software (Calculator) computes Percentiles for any dataset according to 8 different percentile definitions.